Standard Deviation in Roulette

Roulette is among the oldest casino games still in operation. People variously attribute the invention of the game to Blaise Pascal, Italian mathematician Don Pasquale and some others. Anyway, the introduction of the roulette wheel in Paris happened in the year 1765.

The American roulette wheel boasts 38 slots, which feature numbers 00, 0 and 1-36. The slots 0,00 are green while slots 1,3,5,7,9,112,14,16,18,19,21,23,25,27,30,32,34, and 36 are red.

Slots 2,4,6,8,10,11,13,15,17,20,22,24,26,28,29,31,33, and 35 are black. Apart from the 0 and 00 slots, all the other slots alternate between black and red. The designers had one goal when arranging the numbers in a strange way. Their goal was to ensure that the high and low numbers, as well as the even and odd numbers, appear to alternate.

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Roulette experiment is quite simple. The gambler spins the wheel and then rolls a small ball in a groove, in opposite direction as the wheel’s motion. Eventually, the ball drops into a slot. Naturally, gamblers assume that the roulette wheel is fair. Therefore, the random variable X that provides the ball’s slot number remains evenly distributed over the sample.


Space S={00,0,1,…,36}, therefore,
P(X=x)=38 x∈S
P(X=x)=138 for each x∈S

The bets

When it comes to gambling, roulette is a very popular game due to its wider variety of bets the gambler can make. Most bets will appear to have similar expected values – negative value. But the variance will highly differ depending on the particular bet.

All roulette bets share the same expected value. However, the standard deviation will vary inversely with the total number of numbers the gambler selects. What are the significances of that for the gambler?

The theoretical win, the expected value and the standard deviation

The gambler derives the theoretical win form possibilities built into the casino game. Theoretically, all casino games are designed in a certain way to guarantee a good return to the casino, the winnings for the players (the theoretical wins) are always negative while the expectation also known as the expected value (winnings for the particular casino) is usually positive.

But as you expect with any probability, a randomness element exists. Therefore, the theoretical win standard deviation offers a threshold for the casino managers to decide whether the play has passed a limit where it turns suspect.

Theoretical win and the expected value

Gamblers determine the average results using theoretical win formula. A quick example:

The payout of a split bet on single-zero roulette is 17:1. And 2/17 is the possibility of winning the split bet. In other words, 2 numbers are played from a total of 37 numbers. The possibility of losing the wager is 37-2/37=35/37 – for other numbers not played on. The wager theoretical winning formula is:
(Possibility of winning x units of wager payout) – (possibility of losing x bet)

(2/37 x17) – (35/37 x 1) = -0.02703

Alternatively, 50% of every wager on the single-zero roulette split wagers, in the long term; will achieve a $0.02703 loss per every dollar the gambler bets.
The standard deviation

Mostly, results will fall in the range of 3-standard deviations from the mean. To obtain the standard deviation, you have to turn to binomial distribution. The binomial distribution is a statistical dispersion where only 1 of every 2 outcomes is possible. In the above case, the outcome is either a loss or a win.

The binomial standard deviation formula is:

Square Root (the total number of games x the possibility of winning x the possibility of losing) x the average wager x (the unit of wager + the wager units’ payout)
Using the split wager example and $2 over 3 games as the average, the standard deviation formula would be:
Square Root (3 x 2/37 x 35/37 x $2 x (1+17)
= Square Root (3 x 2/37) x $2 x 18
= $14.09972 (Standard deviation of the theoretical win)
The result
Gamblers will use Z-table and the following table to determine whether the win of a gambler is probable.
(Win – Average win (the theoretical win)) / Standard deviation
If the player in the above example won $15, the yield would be:
The gambler wagered 3 games with a mean wager of $2. Therefore, the total bet would be $6
(15 – (-0.02703 x6)) / 14.09972
= 1.075352

The result of the wager is 1.075352, which means that the player’s gaming result is 1.075352-standard deviation from the mean. In other words, the gaming results of the player are in the range of the results 15%.
Importance of standard deviation in roulette casino

Actually, some wagers are more probable than the others are. Therefore, as a wager you will need to look at the probabilities. Probability is a basic ways of measuring the possibility of something happening. It is important in roulette casino game. » For Dummies » Standard Deviation in Roulette
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